Simplify and expand the following expression: $ \dfrac{a}{2a + 7}+\dfrac{4a + 1}{a - 9} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2a + 7)(a - 9)$ Multiply the first term by $\dfrac{a - 9}{a - 9}$ $ \begin{align*} \dfrac{a}{2a + 7} \times \dfrac{a - 9}{a - 9} & = \dfrac{(a)(a - 9)}{(2a + 7)(a - 9)} \\ & = \dfrac{a^2 - 9a}{(2a + 7)(a - 9)}\end{align*} $ Multiply the second term by $\dfrac{2a + 7}{2a + 7}$ $ \begin{align*} \dfrac{4a + 1}{a - 9} \times \dfrac{2a + 7}{2a + 7} & = \dfrac{(4a + 1)(2a + 7)}{(a - 9)(2a + 7)} \\ & = \dfrac{8a^2 + 30a + 7}{(a - 9)(2a + 7)}\end{align*} $ Now we have: $ = \dfrac{a^2 - 9a}{(2a + 7)(a - 9)} + \dfrac{8a^2 + 30a + 7}{(a - 9)(2a + 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{a^2 - 9a + 8a^2 + 30a + 7}{(2a + 7)(a - 9)} $ $ = \dfrac{9a^2 + 21a + 7}{(2a + 7)(a - 9)}$ Expand the denominator: $ = \dfrac{9a^2 + 21a + 7}{2a^2 - 11a - 63}$